Question

which of the following statements are true? check all of the boxes that apply.

(sin a=\frac{h}{c})

(b sin a=a sin b)

(\frac{a}{sin a}=\frac{b}{sin b})

(\frac{b}{sin b}=\frac{c}{sin c})

Explanation:

Step1: Recall the sine definition

In a right - triangle (the right - triangle formed by \(h\), part of \(b\) and \(c\)), \(\sin A=\frac{h}{c}\) (by the definition of sine: \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\)).

Step2: Use the sine formula in two right - triangles

In the right - triangle with height \(h\) and side \(b\), \(h = b\sin A\). In the right - triangle with height \(h\) and side \(a\), \(h=a\sin C\) (if we consider the other non - right angle). But if we use the law of sines \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Cross - multiplying \(\frac{a}{\sin A}=\frac{b}{\sin B}\) gives \(a\sin B = b\sin A\).

Step3: Analyze each option

• For \(\sin A=\frac{h}{c}\): True, by the definition of sine in the right - triangle with hypotenuse \(c\) and opposite side \(h\) relative to angle \(A\).
• For \(b\sin A=a\sin B\): True, from the law of sines \(\frac{a}{\sin A}=\frac{b}{\sin B}\) (cross - multiply \(a\sin B=b\sin A\)).
• For \(\frac{a}{\sin A}=\frac{b}{\sin B}\): True, this is the law of sines.
• For \(\frac{b}{\sin B}=\frac{c}{\sin C}\): There is no information in the given diagram (with height \(h\) drawn from \(B\) to \(AC\)) that directly supports this. The law of sines \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\) is a general formula for any triangle, but based on the construction of height \(h\) from \(B\) (not from \(A\) or \(C\)), we cannot directly derive \(\frac{b}{\sin B}=\frac{c}{\sin C}\) from the right - triangle relationships shown in the diagram.

Answer:

\(\sin A=\frac{h}{c}\), \(b\sin A = a\sin B\), \(\frac{a}{\sin A}=\frac{b}{\sin B}\)